MODULUS AND DAMPING OF SOILS DUE TO CYCLIC SHEARl

by B. 0. Hardin

Professor of Civil Engineering and

V. P. Drnevich Assistant Professor of Civil Engineering

University of Kentucky

The aims of this research are to determine the shear modulus and damping of soils as functions of ambient confining stresses, soil type, void ratio, cycles of loading, strain amplitude, and frequency. Both undisturbed and disturbed samples are being tested. Soils range from the sensitive Leda clay of Canada to San Francisco sand .

Three different test apparatus were used. The first applied a pseudo-static cyclic shear stress (1 / 12 cycle per second) to a hollow cylindrical soil specimen. The second applied dynamic cyclic shear stress (30 to 100 cycles per second) to a hollow cylindrical specimen. The third applied dynamic cyclic shear stress (200 to 300 cycles per second) to triaxial type specimens.

As shearing strain amplitude approaches zero, the shear modulus approaches a maximum value which is given by

G max= 1230 (OCR) K F (e) u;, ~

where: G max is shear modulus at strains approaching zero, OCR is the over-consolidation ratio, K is a parameter related to the plasticity index, F(e) is given by (2 .973-e) 2 / (1 + e), e is the void ratio, and o;; is the mean effective principal stress.

The shear moduli at larger strain amplitudes are related to G max , the shearing strain amplitude, and the shear strength of the soil. The relationship is in the modified hyperbolic form .

For damping, a term D max, the maximum damping ratio for a given soil can be determined. Because damping increases with shearing strain amplitude, D max occurs at large amplitudes. The empirical equation for D max is in the process of final preparation.

Based on data frqm many soils, it appears that damping and modulus for a soil can be related by the simple expression

GI G max= 1-(o ID maJ

where G and D are the shear modulus at a given-i;Lrnin amplitude.

The more sophisticated analytical techniques such as viscoelasticity, finite elements, etc . are limited by the soil data used. With accurate modulus and damping data, these techniques can become useful for such soil-structure interaction problems as earthquake response, machine foundations, pavement behavior, and vibration isolation.

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Figure 3. Static Hollow-Cylinder Shear Apparatus Figure 4. Hollow-Cylinder Resonant Shear Apparatus

1.0

0.8

0.6 G

G max 0.4

0.2

0 0

MODULUS CURVES

Clean Dry Sand

Cohesive Soils

I upper limit 2 probable value 3 lower limit

pro a le value

5 er;,= 7 kg I cm2 6 CTo: 2 kg /cm2 7 'To= 0.5 kg./ cm2

Cohesive soil curves approach curve 3 for very high pressures. These curves do not include effects of dilational inertia or vibration pore pressures.

2 3

Figure 5. Normalized Shear Modulus versus Normalized Shear Strain for Various Soils

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20

DAMPING

Clean Dry Sands

Cohesive Soils

All Soils

CURVES

I 2

lower limit average value

3 lower limit

5 o-o = 2 kg/ cm2 Average 4 CTo : 7 kg/cm2 }

6 CTo = 0.5 kg /cm2 Values

7 upper limit

Note : The damping for the f irst cycle of loading or for very low pressures (less than 0. 25 kg/cm2) may exceed in a few cases, the limit defined by curve 7 . Similarly , fo r pressures greater than 7. 0 kg/cm2 and for a large number of cycles, the damping may in a few cases be lower than the limits defined b~y~cur~ v'.:e~s_21~a~n~d~3i.'-""-------------------1

D %

i . , , " ~ " . . .c

"'

10

O-=::....~~~~~~~~~-'-~~~~~~~~~-L.~~~~~~~~~--'

0

1200

1000

800

600

400

200

2

Figure 6. Damping Ratio versus Normalized Strain

Silica Sand Rel. Density = 70 . Conf . Press .

2=

1.0 kg/cm

Silica Sand

20 Rel. Density "' 70% Conf . Press .... 1. 0 kg/cm2

16

... c 12

~ ~-~-~-~~10--1~2-~14,----1~6-~18,----:-'.20 x 10- 4 0 0 10 12 14 16 18

She a ring Strni n Ampli tude Shearing Strain Amplitude

Figure 7. Shear Modulus and Damping versus Shearing Strain for Dry and Saturated Sands

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3